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  <h1 id="数学-高等数学 第17讲 空间几何" class="content-subhead">数学-高等数学 第17讲 空间几何</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学-高等数学 8 第17讲 空间几何.html" class="leancloud_visitors" style="display:none" data-flag-title="数学-高等数学 第17讲 空间几何"></span>
  </p>
  <h2 id="17">第17讲 空间几何</h2>
<h3 id="1">1. 向量基础知识</h3>
<h4 id="1_1">（1）数量积（内积，点乘）</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} \cdot \overrightarrow{\pmb{b}} 
= \bigg|\overrightarrow{\pmb{a}}\bigg|\cdot\bigg|\overrightarrow{\pmb{b}}\bigg|\cos\theta
</script>
</p>
<h4 id="2">（2）向量积（外积，叉乘）</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} \times \overrightarrow{\pmb{b}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{vmatrix}
</script>
</p>
<h4 id="3">（3）混合积</h4>
<p>
<script type="math/tex; mode=display">
[\pmb{a}\pmb{b}\pmb{c}] =
\pmb{a} \times \pmb{b} · \pmb{c} = 
\begin{vmatrix}
a_x & a_y & a_z \\
b_x & b_y & b_z \\
c_x & c_y & c_z
\end{vmatrix}
</script>
</p>
<h4 id="4">（4）方向角</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}} 与\ x,y,z\ 轴的夹角\ \alpha,\beta,\gamma
</script>
</p>
<h4 id="5">（5）方向余弦</h4>
<p>
<script type="math/tex; mode=display">
\cos\alpha = \cfrac{a_x}{|\overrightarrow{\pmb{a}}|}, \ \ \ 
\cos\beta  = \cfrac{a_y}{|\overrightarrow{\pmb{a}}|}, \ \ \
\cos\gamma = \cfrac{a_z}{|\overrightarrow{\pmb{a}}|}
</script>
</p>
<h4 id="6">（6）方向单位向量</h4>
<p>
<script type="math/tex; mode=display">
\overrightarrow{\pmb{a}^o} = \cfrac{\overrightarrow{\pmb{a}}}{|\overrightarrow{\pmb{a}}|} = (\cos\alpha,\cos\beta,\cos\gamma)
</script>
</p>
<h3 id="2_1">2. 空间直线与空间平面</h3>
<h4 id="1_2">（1）空间直线的方向向量与法平面</h4>
<h5 id="1_3">1、一般式</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
A_1 x + B_1 y + C_1 z + D_1 = 0 \\[2ex]
A_2 x + B_2 y + C_2 z + D_2 = 0
\end{cases}
</script>
</p>
<h5 id="2_2">2、标准式（点向式，对称式）</h5>
<p>
<script type="math/tex; mode=display">
直线过定点：P(x_0,y_0,z_0) \\[1ex]
直线的方向向量：\overrightarrow{l} = (m,n,p) \\[2ex]
\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} \\[1em]
(m,n,p的值为0，意味着对应的分子为0)
</script>
</p>
<h5 id="3_1">3、参数式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} = t \\[1em]
\begin{cases}
x = x_0 + mt \\[2ex]
y = y_0 + nt \\[2ex]
z = z_0 + pt
\end{cases}
</script>
</p>
<h5 id="4_1">4、两点式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x-x_1}{x_2-x_1} = \cfrac{y-y_1}{y_2-y_1} = \cfrac{z-z_1}{z_2-z_1}
</script>
</p>
<h4 id="2_3">（2）空间平面与法向量</h4>
<h5 id="1_4">1、一般式方程</h5>
<p>
<script type="math/tex; mode=display">
A x + B y + C z + D = 0
</script>
</p>
<h5 id="2_4">2、点法式方程</h5>
<p>
<script type="math/tex; mode=display">
平面过定点：P(x_0,y_0,z_0) \\[1ex]
平面法向量：\overrightarrow{n} = (A,B,C) \\[2ex]
A(x-x_0) + B(y-y_0) + C(z-z_0) = 0
</script>
</p>
<h5 id="3_2">3、三点式</h5>
<p>
<script type="math/tex; mode=display">
\begin{vmatrix}
x-x_0 & y-y_0 & z-z_0 \\
x-x_1 & y-y_1 & z-z_1 \\
x-x_2 & y-y_2 & z-z_2
\end{vmatrix} = 0
</script>
</p>
<h5 id="4_2">4、截距式</h5>
<p>
<script type="math/tex; mode=display">
\cfrac{x}{a}+\cfrac{y}{b}+\cfrac{z}{c} = 1
</script>
</p>
<h4 id="3_3">（3）平面与直线的关系</h4>
<h5 id="_1">直线之间的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
l_1直线方向向量：\overrightarrow{l_1} = (m_1,n_1,p_1) \\[1ex]
l_2直线方向向量：\overrightarrow{l_2} = (m_2,n_2,p_2) \\[3ex]
\cos\theta = \cfrac{\bigg|\overrightarrow{l_1}·\overrightarrow{l_2}\bigg|}{\bigg|\overrightarrow{l_1}\bigg|\bigg|\overrightarrow{l_2}\bigg|}
</script>
</p>
<h4 id="4_3">（4）平面与平面的关系</h4>
<p>
<script type="math/tex; mode=display">
\pi_1：A_1 x + B_1 y + C_1 z + D_1 = 0 \\[1ex]
\pi_2：A_2 x + B_2 y + C_2 z + D_2 = 0
</script>
</p>
<h5 id="1_5">1、平面法向量间的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
\pi_1平面法向量：\overrightarrow{n_1} = (A_1,B_1,C_1) \\[1ex]
\pi_2平面法向量：\overrightarrow{n_2} = (A_2,B_2,C_2) \\[3ex]
\cos\theta = \cfrac{|\overrightarrow{n_1}·\overrightarrow{n_2}|}{|\overrightarrow{n_1}||\overrightarrow{n_2}|}
</script>
</p>
<h5 id="2-px_0y_0z_0">2、点 <script type="math/tex">P(x_0,y_0,z_0)</script> 到平面的距离</h5>
<p>
<script type="math/tex; mode=display">
d = \cfrac{|Ax_0+By_0+Cz_0+D|}{\sqrt{A^2+B^2+C^2}}
</script>
</p>
<h5 id="3_4">3、任意两个 <strong>平行</strong> 平面之间的距离</h5>
<p>
<script type="math/tex; mode=display">
d = \cfrac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}
</script>
</p>
<h4 id="5_1">（5）直线与平面的关系</h4>
<h5 id="1_6">1、直线方向向量与平面法向量的夹角（锐角/直角）</h5>
<p>
<script type="math/tex; mode=display">
l直线方向向量：\overrightarrow{l} = (m,n,p) \\[1ex]
\pi平面法向量：\overrightarrow{n} = (A,B,C) \\[3ex]
\sin\theta = \cfrac{\bigg|\overrightarrow{l}·\overrightarrow{n}\bigg|}{\bigg|\overrightarrow{l}\bigg|\bigg|\overrightarrow{n}\bigg|}
</script>
</p>
<h5 id="2-l">2、过直线 <script type="math/tex">l</script> 的平面束方程</h5>
<p>
<script type="math/tex; mode=display">
直线\ l\ 由
\begin{cases}
A_1 x + B_1 y + C_1 z + D_1 = 0 \\[2ex]
A_2 x + B_2 y + C_2 z + D_2 = 0
\end{cases}
确定
</script>
</p>
<p>平面束方程写法1<br />
<script type="math/tex; mode=display">
过\ l\ 的平面束：(A_1 x + B_1 y + C_1 z + D_1) + \lambda(A_2 x + B_2 y + C_2 z + D_2) = 0 \\[1ex]
不包含平面：A_2 x + B_2 y + C_2 z + D_2 = 0
</script>
<br />
平面束方程写法2<br />
<script type="math/tex; mode=display">
过\ l\ 的平面束：\lambda_1(A_1 x + B_1 y + C_1 z + D_1) + \lambda_2(A_2 x + B_2 y + C_2 z + D_2) = 0 \\[1ex]
\lambda_1^2+\lambda_2^2 \neq 0 \\[1ex]
包含所有平面
</script>
</p>
<h3 id="3_5">3. 空间曲线和空间曲面</h3>
<h4 id="1_7">（1）空间曲线的切线与法平面</h4>
<h5 id="1_8">1、参数方程给出曲线</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
x = x(t) \\[2ex]
y = y(t) \\[2ex]
z = z(t) 
\end{cases}
</script>
</p>
<p>曲线在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的切向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} = (x'(t_0), y'(t_0), z'(t_0))
</script>
</p>
<h5 id="2_5">2、用方程组给出曲线</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}
确定
\begin{cases}
x = x \\[2ex]
y = y(x) \\[2ex]
z = z(x) 
\end{cases}
</script>
</p>
<p>曲线在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的切向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} = (1, y'(x_0), z'(x_0))
</script>
</p>
<h4 id="2_6">（2）空间曲面的切平面与法线</h4>
<h5 id="1_9">1、隐式给出曲面</h5>
<p>
<script type="math/tex; mode=display">
F(x,y,z)=0
</script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = (F'_x|_{P_0}, F'_y|_{P_0}, F'_z|_{P_0})
</script>
</p>
<h5 id="2_7">2、显式给出曲面</h5>
<p>
<script type="math/tex; mode=display">
z=z(x,y)
</script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = (z'_x(x_0,y_0), z'_y(x_0,y_0), -1)
</script>
</p>
<h5 id="3_6">3、用参数方程给出曲面</h5>
<p>
<script type="math/tex; mode=display">
\begin{cases}
x = x(u,v) \\[2ex]
y = y(u,v) \\[2ex]
z = z(u,v) 
\end{cases}
</script>
</p>
<p>固定 <script type="math/tex">v=v_0</script> ，得到 <script type="math/tex">u </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{\pmb{l}_1} = (x'_u, y'_u, z'_u)|_{P_0} </script>
</p>
<p>固定 <script type="math/tex">u=u_0</script> ，得到 <script type="math/tex"> v </script> 曲线在 <script type="math/tex"> P_0 </script> 处的切向量为 <script type="math/tex"> \overrightarrow{\pmb{l}_2} = (x'_v, y'_v, z'_v)|_{P_0} </script>
</p>
<p>曲面在点 <script type="math/tex"> P_0(x_0,y_0,z_0) </script> 的法向量<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{n}} = \overrightarrow{\pmb{l_1}} \times \overrightarrow{\pmb{l_2}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
l_{1x} & l_{1y} & l_{1z} \\
l_{2x} & l_{2y} & l_{2z}
\end{vmatrix}
</script>
</p>
<blockquote class="content-quote">
<p>曲线 <script type="math/tex">\rightarrow</script> 切向量 <script type="math/tex">\rightarrow</script> 切线 <script type="math/tex">\rightarrow</script> 法平面</p>
<p>曲面 <script type="math/tex">\rightarrow</script> 法向量 <script type="math/tex">\rightarrow</script> 法线 <script type="math/tex">\rightarrow</script> 切平面<br />
<script type="math/tex; mode=display">
\overrightarrow{\pmb{l}} \ \ \  or\ \ \ \overrightarrow{\pmb{n}} = (A, B, C) \\[1em]
\cfrac{x-x_0}{A} = \cfrac{y-y_0}{B} = \cfrac{z-z_0}{C} \\[1em]
A(x-x_0)+B(y-y_0)+C(z-z_0)=0
</script>
</p>
</blockquote>
<h3 id="4_4">4. 空间曲线在坐标面上的投影</h3>
<p>
<script type="math/tex; mode=display">
曲线\ \Gamma：
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}
在\ xOy\ 平面上的投影包含于曲线
\begin{cases}
\varphi(x,y)=0 \\[2ex]
z=0
\end{cases}
</script>
</p>
<h3 id="5_2">5. 旋转曲面</h3>
<h4 id="1_10">（1）曲线绕任意直线旋转</h4>
<p>
<script type="math/tex; mode=display">
曲线\ \Gamma：
\begin{cases}
F(x,y,z)=0 \\[2ex]
G(x,y,z)=0
\end{cases}\ 
绕直线\ 
L：\cfrac{x-x_0}{m} = \cfrac{y-y_0}{n} = \cfrac{z-z_0}{p} \ 
旋转
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学-高等数学 第17讲 空间几何.assets/IMG_EFCEA317AD4E-1.jpeg" alt="IMG_EFCEA317AD4E-1" style="zoom:25%;" /><br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
\overrightarrow{M_1P}\ \bot\ \overrightarrow{\pmb{s}} \\[2ex]
\bigg|\overrightarrow{M_0P}\bigg| = \bigg|\overrightarrow{M_0M_1}\bigg|
\end{cases} \\[2ex]
\Rightarrow
&\begin{cases}
m(x-x_1)+n(y-y_1)+p(z-z_1) = 0 \\[2ex]
(x-x_0)^2+(y-y_0)^2+(z-z_0)^2 = (x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2
\end{cases} \\[2ex]
且
&\begin{cases}
F(x_1,y_1,z_1)=0 \\[2ex]
G(x_1,y_1,z_1)=0
\end{cases} \\[2ex]
&消去\ x_1,y_1,z_1\ 后得到\ x,y,z\ 的表达式
\end{split}\end{equation}
</script>
</p>
<h4 id="2_8">（2）曲线绕坐标轴旋转</h4>
<p>解法1<br />
<script type="math/tex; mode=display">
绕\ x\ 轴：x=x，y=z=\pm\sqrt{y^2+z^2} \\[1ex]
绕\ y\ 轴：y=y，z=x=\pm\sqrt{z^2+x^2} \\[1ex]
绕\ z\ 轴：z=z，x=y=\pm\sqrt{x^2+y^2} \\[1ex]
绕哪个轴，哪个轴不变，其他两个轴等于另外两个轴各自的平方之和再开根号
</script>
<br />
解法2<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
&\begin{cases}
F(x_1,y_1,z_1)=0 \\[2ex]
G(x_1,y_1,z_1)=0 \\[2ex]
x^2+y^2=x_1^2+y_1^2
\end{cases} \\[2ex]
&消去\ x_1,y_1,z_1\ 后得到\ x,y,z\ 的表达式
\end{split}\end{equation}
</script>
</p>
<h3 id="6_1">6. 场论</h3>
<h4 id="1_11">（1）方向导数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial\overrightarrow{\pmb{l}}} \bigg|_{P_0}
&= \lim_{t \to 0^+}\frac{u(x_0 + \Delta x, y_0 + \Delta y, z_0 + \Delta z) - u(x_0, y_0, z_0)}{t} \\[2ex]
&= \lim_{t \to 0^+}\frac{u(x_0 + t\cos\alpha, y_0 + t\cos\beta, z_0 + t\cos\gamma) - u(x_0, y_0, z_0)}{t} \\[3ex]
(若u可微)
&= \lim_{t \to 0^+}\frac{u'_x(P_0)\Delta x + u'_y(P_0)\Delta y + u'_z(P_0)\Delta z - o(t)}{t} \\[3ex]
&= u'_x(P_0)\cos\alpha + u'_y(P_0)\cos\beta + u'_z(P_0)\cos\gamma \\[3em]
其中\ \overrightarrow{\pmb{l}^o}&=(\cos\alpha,\cos\beta,\cos\gamma)\ 为单位向量 \\[2ex]
t &= \sqrt{(\Delta x)^2+(\Delta y)^2+(\Delta z)^2}\\[3em]
\end{split}\end{equation}
</script>
</p>
<h5 id="_2">方向导数与偏导数的关系</h5>
<p>偏导数的定义：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial x}
&= \lim_{\Delta x \to 0}\frac{u(x_0 + \Delta x, y_0, z_0) - u(x_0, y_0, z_0)}{\Delta x} = u_x' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(1,0,0)\ 即 \ x\ 轴正方向的方向导数\\[2ex]
\cfrac{\partial u}{\partial y}
&= \lim_{\Delta y \to 0}\frac{u(x_0, y_0 + \Delta y, z_0) - u(x_0, y_0, z_0)}{\Delta y} = u_y' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(0,1,0)\ 即 \ y\ 轴正方向的方向导数\\[2ex]
\cfrac{\partial u}{\partial z}
&= \lim_{\Delta z \to 0}\frac{u(x_0, y_0, z_0 + \Delta z) - u(x_0, y_0, z_0)}{\Delta z} = u_z' \ \ \ 沿\ \overrightarrow{\pmb{l}}=(0,0,1)\ 即 \ z\ 轴正方向的方向导数\\[2ex]
\end{split}\end{equation}
</script>
</p>
<h4 id="2_9">（2）梯度</h4>
<p>若 <script type="math/tex">u(x,y,z)</script> 具有一阶偏导数<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\overrightarrow{\mathbf{grad}}\ u \bigg|_{P_0}
&= \cfrac{\partial u}{\partial x}\ \overrightarrow{\pmb{i}} + \cfrac{\partial u}{\partial y}\ \overrightarrow{\pmb{j}} + \cfrac{\partial u}{\partial z}\ \overrightarrow{\pmb{k}} \\[3ex]
&= (u'_x(P_0), u'_y(P_0), u'_z(P_0))
\end{split}\end{equation}
</script>
</p>
<h5 id="_3">方向导数与梯度的关系</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\cfrac{\partial u}{\partial \overrightarrow{\pmb{l}}} \bigg|_{P_0}
&= u'_x\cos\alpha + u'_y\cos\beta + u'_z\cos\gamma \\
&= (u'_x, u'_y, u'_z) · (\cos\alpha,\cos\beta,\cos\gamma) \\[2ex]
&= \overrightarrow{\mathbf{grad}}\ u \big|_{P_0}· \overrightarrow{\pmb{l}^o} \\[2ex]
&= \bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|·\bigg|\ \overrightarrow{\pmb{l}^o}\ \bigg|\cos\theta \\[2ex]
&= \bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|\cos\theta\\[1em]
\end{split}\end{equation}
</script>
</p>
<ul>
<li>由于 <script type="math/tex">\bigg|\ \overrightarrow{\pmb{l}}\ \bigg|=1</script> ，<strong>梯度的方向</strong> 即 <strong>取得最大方向导数的方向</strong></li>
<li>最大方向导数为 <script type="math/tex">\bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|</script>，即 <strong>梯度的模</strong></li>
<li>最小方向导数为 <script type="math/tex">-\bigg|\overrightarrow{\mathbf{grad}}\ u\big|_{P_0}\bigg|</script>
</li>
</ul>
<h4 id="3_7">（3）散度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{\pmb{A}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}} </script> ，则<br />
<script type="math/tex; mode=display">
div\ \overrightarrow{\pmb{A}} = \cfrac{\partial P}{\partial x} + \cfrac{\partial Q}{\partial y} + \cfrac{\partial R}{\partial z}
</script>
</p>
<ul>
<li>
<p>
<script type="math/tex"> div\ \overrightarrow{\pmb{A}} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处源头的强弱程度。</p>
</li>
<li>
<p>若  <script type="math/tex"> div\ \overrightarrow{\pmb{A}} = 0 </script>  在场内处处成立，则称A为<strong>无源场</strong>。</p>
</li>
</ul>
<h4 id="4_5">（4）旋度</h4>
<p>向量场 <script type="math/tex"> \overrightarrow{\pmb{A}}(x,y,z) = P(x,y,z)\overrightarrow{\pmb{i}}+Q(x,y,z)\overrightarrow{\pmb{j}}+R(x,y,z)\overrightarrow{\pmb{k}} </script> ，则<br />
<script type="math/tex; mode=display">
\overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} = 
\begin{vmatrix}
\overrightarrow{\pmb{i}} & \overrightarrow{\pmb{j}} & \overrightarrow{\pmb{k}} \\
\cfrac{\partial}{\partial x} & \cfrac{\partial}{\partial y} & \cfrac{\partial}{\partial z} \\
P & Q & R
\end{vmatrix}
</script>
</p>
<ul>
<li>
<script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} </script>  表示场在 <script type="math/tex"> (x,y,z) </script> 处最大旋转趋势的度量。</li>
<li>若  <script type="math/tex"> \overrightarrow{\mathbf{rot}}\ \overrightarrow{\pmb{A}} = 0 </script>  在场内处处成立，则称A为<strong>无旋场</strong>。</li>
</ul>
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